 Solution

A symmetry-closed orbital basis, often called a complete basis in this context, is a linearly independent set whose span is carried into itself by every group operation. No transformed basis function requires a function outside that span. This is completeness for the chosen group representation, not a claim that a small atomic-orbital set spans the full electronic Hilbert space.

A representation matrix $D(g)$ specifies the linear transformation produced by a molecular symmetry operation $g$. With the column convention,Its character of a representation is the matrix trace,The character is unchanged by a simultaneous change of basis and is constant on a conjugacy class, which is why character tables have one column per class.